202 research outputs found
Isomorphic properties of Intersection bodies
We study isomorphic properties of two generalizations of intersection bodies,
the class of k-intersection bodies and the class of generalized k-intersection
bodies. We also show that the Banach-Mazur distance of the k-intersection body
of a convex body, when it exists and it is convex, with the Euclidean ball, is
bounded by a constant depending only on k, generalizing a well-known result of
Hensley and Borell. We conclude by giving some volumetric estimates for
k-intersection bodies
Complex Intersection Bodies
We introduce complex intersection bodies and show that their properties and
applications are similar to those of their real counterparts. In particular, we
generalize Busemann's theorem to the complex case by proving that complex
intersection bodies of symmetric complex convex bodies are also convex. Other
results include stability in the complex Busemann-Petty problem for arbitrary
measures and the corresponding hyperplane inequality for measures of complex
intersection bodies
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